We study online algorithms for maximum cardinality matchings with edge arrivals in graphs of low degree. Buchbinder, Segev, and Tkach showed that no online algorithm for maximum cardinality fractional matchings can achieve a competitive ratio larger than $4/(9-\sqrt 5)\approx 0.5914$ even for graphs of maximum degree three. The negative result of Buchbinder et al. holds even when the graph is bipartite and edges are revealed according to vertex arrivals, i.e. once a vertex arrives, all edges are revealed that include the newly arrived vertex and one of the previously arrived vertices. In this work, we complement the negative result of Buchbinder et al. by providing an online algorithm for maximum cardinality fractional matchings with a competitive ratio at least $4/(9-\sqrt 5)\approx 0.5914$ for graphs of maximum degree three. We also demonstrate that no online algorithm for maximum cardinality integral matchings can have the competitive guarantee $0.5807$, establishing a gap between integral and fractional matchings for graphs of maximum degree three. Note that the work of Buchbinder et al. shows that for graphs of maximum degree two, there is no such gap between fractional and integral matchings, because for both of them the best achievable competitive ratio is $2/3$. Also, our results demonstrate that for graphs of maximum degree three best possible competitive ratios for fractional matchings are the same in the vertex arrival and in the edge arrival models.

翻译：本文研究低度图中边到达情形下最大基数匹配的在线算法。Buchbinder、Segev与Tkach证明，即使在最大度三的图中，最大基数分数匹配的在线算法竞争比也无法超过$4/(9-\sqrt 5)\approx 0.5914$。Buchbinder等人的否定性结论甚至在二分图且边按顶点到达方式呈现时依然成立，即当新顶点到达时，所有连接该顶点与已到达顶点的边会同时呈现。本工作通过构造一个竞争比至少为$4/(9-\sqrt 5)\approx 0.5914$的最大基数分数匹配在线算法，补充了Buchbinder等人的否定性结果。我们进一步证明最大基数整数匹配的在线算法竞争比不可能达到$0.5807$，从而在最大度三图中确立了分数匹配与整数匹配之间的性能差距。值得注意的是，Buchbinder等人的研究表明在最大度二图中不存在此类差距，因为分数匹配与整数匹配均可达到的最佳竞争比均为$2/3$。此外，我们的结果证明对于最大度三图，分数匹配在顶点到达模型与边到达模型中能达到的最佳竞争比是相同的。